Improved model-free adaptive control method

ABSTRACT

The present invention discloses an improved model-free adaptive control method, in particular to an improved method for a compact dynamic linearization model-free adaptive control based on MIMO systems, and belongs to the field of control algorithm design. Firstly, proportional control is added in CFDL-MFAC to improve the problems of low response speed and large overshoot in the original control system. Secondly, an anti-windup control algorithm of an actuator is added in the above control structure so that the actuator does not conduct transfinite operation when reaching the upper or lower saturation limit and the actuator can quickly make a control response when a control instruction enters an unsaturated region again to improve the control accuracy of the system. Then, it is proved through strict analysis that the improved control algorithm can ensure tracking error and BIBO stability under certain conditions. Finally, the above control algorithm is applied to an aero-engine control system, and the effectiveness and superiority of the above control algorithm can be obtained by numerical experiments.

TECHNICAL FIELD

The present invention discloses an improved method for a compact dynamiclinearization model-free adaptive control based on Multiple-InputMultiple-Output (MIMO) systems, and belongs to the field of controlalgorithm design.

BACKGROUND

With the technological innovation and industrial progress, people havehigher requirements for the safety, stability and efficient control ofthe aircraft, and the model-based control concept is greatly affected bythe modeling accuracy. In addition, the model obtained by mathematicalmodeling will gradually deviate from the actual control object due tounmodeled dynamics problems and engine mechanical wear. These controlblind spots cause the control effects to be worse with the extension ofuse time. Therefore, a model-free adaptive control (MFAC) algorithmemerges at the right moment.

MFAC is a data-driven control method. And the parameter design does notdepend on the structure of a control object, that is, it does not needto model the controlled object or identify the parameters, but onlydesigns the control parameters through the input and output data of thecontrol system. The method was first proposed by Hou Zhongsheng,including the new dynamic linearization method and the concept of apseudo Jacobian matrix (PJM). The pseudo Jacobian matrix can be directlyestimated from the input and output data. In the past two decades, themethod has obtained important research results in theory andapplication. Studies have shown that the MFAC method is easier to beused and has better control effect than other DDC methods, such as IFTand VRFT.

Recently, many literatures have mentioned the extended research andapplication of the MFAC algorithm, such as adaptive iterative learningcontrol, adaptive online learning control and model-free adaptivedecoupling control. In addition, in the past five years, the researchupsurge of combining the MFAC algorithm with iterative learning hasgradually increased. For example, the adaptive iterative learningcontrol method is used to solve the problem of macro highway trafficdensity control of ramp control, the problem of random packet loss andthe problems of parking control and tracking control of train tracks inrailway stations. In addition, the combination of the neural networkwith MFAC has also been applied. However, most of the previousaero-engine control studies focus on model-based control. Therefore, itis of important practical significance to extend MFAC to the field ofaero-engine control. The MFAC control strategy uses the input and outputdata of the system to update the PJM in real time through a parameterestimation algorithm, so that the controller parameters can be updatedin real time to allow the controller to carry out timely and stablecontrol on the aircraft in case of serious change in the flightenvironment, so as to ensure safe flight of the aircraft at differentflight heights and atmospheric environment.

SUMMARY

In view of the deficiencies of the existing compact dynamiclinearization model-free adaptive control method in discrete simulationapplication of complex models, the present invention proposes animproved method for a compact dynamic linearization model-free adaptivecontrol based on MIMO systems, which is suitable for the design andapplication fields of control systems and can be used for improving theperformance of the control systems and mainly solving the problems oflow response speed, large overshoot and actuator saturation in themodel-free adaptive control method.

The technical solution of the present invention is as follows:

An improved method for a compact dynamic linearization model-freeadaptive control (CFDL-MFAC) based on MIMO systems comprises thefollowing steps:

step A: analyzing the existing method for the compact dynamiclinearization model-free adaptive control, and from experimentalresults, finding that the application process has deficiencies inresponse time and stability;

the MIMO discrete-time nonlinear system is expressed as follows.

y(k+1)=f(y(k), . . . ,y(k−n _(y)),u(k), . . . ,u(k−n _(u)))  (1)

wherein u(k) and y(k) are system inputs and system outputs at time k,respectively; n_(y) and n_(u) are two unknown integers; f( . . . )=(f₁(. . . ), . . . , f_(m)( . . . )) is an unknown nonlinear function;

when f has a continuous partial derivative condition and formula (1)satisfies a generalized Lipschitz condition, expressing formula (1) asthe following CFDL data model form:

$\begin{matrix}{{{\Delta\;{y\left( {k + 1} \right)}} = {{{\Phi_{c}(k)}\Delta\;{u(k)}\mspace{14mu}{wherein}\mspace{14mu}{\Phi_{c}(k)}} = {\begin{bmatrix}{\phi_{11}(k)} & {\phi_{12}(k)} & \ldots & {\phi_{1\; m}(k)} \\{\phi_{21}(k)} & {\phi_{22}(k)} & \ldots & {\phi_{2\; m}(k)} \\\vdots & \vdots & \vdots & \vdots \\{\phi_{m\; 1}(k)} & {\phi_{m\; 2}(k)} & \ldots & {\phi_{mm}(k)}\end{bmatrix} \in R^{m \times m}}}};} & (2)\end{matrix}$

firstly, proposing the following assumptions:

assumption 1: Φ_(c)(k) as a pseudo Jacobian matrix of the system shallbe a diagonal dominant matrix which satisfies the following conditions:|ϕ_(ij)|≤b₁,b₂≤|_(ii)(k)|≤αb₂,α≥1,b₂>b₁(2α+1)(m−1), i=1, . . . ,m, j=1,. . . ,m, i≠j; b₁ and b₂ are set as bounded constants, i and j are setas row and column indexes of the matrix; and the signs of all elementsin Φ_(c)(k) remain the same at any time k;

expressing a control input criterion function as formula (3):

J(u(k))=∥y*(k+1)−y(k+1)∥² +λ∥u(k)−u(k−1)∥²  (3)

wherein λ>0 represents a weight factor, which is used to punish thechange of excessive control input quantity; y*(k+1) is a desired outputsignal;

substituting formula (2) into formula (3), deriving u(k) and making theequation equal to zero to obtain a control input algorithm as follows:

$\begin{matrix}{{u(k)} = {{u\left( {k - 1} \right)} + \frac{{{\rho\Phi}_{c}^{T}(k)}\left( {{y^{*}\left( {k + 1} \right)} - {y(k)}} \right)}{\lambda + {{\Phi_{c}(k)}}^{2}}}} & (4)\end{matrix}$

considering the following parameter estimation criteria function:

J(Φ_(c)(k))=∥Δy(k)−Φ_(c)(k)Δu(k−1)∥²+μ∥Φ_(c)(k)−{circumflex over(Φ)}_(c)(k−1)∥²  (5)

wherein μ is a weight factor used to punish excessive changes in PJMestimates; {circumflex over (Φ)}_(c)(k) is an estimate of Φ_(c)(k);

deriving Φ_(c)(k) in formula (5) and making the equation equal to zeroto obtain a parameter estimation algorithm as follows:

$\begin{matrix}{{{\hat{\Phi}}_{c}(k)} = {{{\hat{\Phi}}_{c}\left( {k - 1} \right)} + \frac{{\eta\left( {{\Delta\;{y(k)}} - {{{\hat{\Phi}}_{c}\left( {k - 1} \right)}\Delta\;{u\left( {k - 1} \right)}}} \right)}\Delta\;{u^{T}\left( {k - 1} \right)}}{\mu + {{\Delta\;{u\left( {k - 1} \right)}}}^{2}}}} & (6)\end{matrix}$

conducting parameter estimation in each k by the above control parameterestimation algorithm to provide control inputs at the time; however, thecalculation of the parameter estimation algorithm needs to occupy acertain time, causing slow system response and causing the controlalgorithm to be limited in use for a system with a small requirement fora control period; and the system vibrates greatly under non-idealconditions from the experimental results;

step B: based on the above problems of slow response and vibration,considering the following improved solution;

Δu(k)=Δu _(m)(k)′+Δu _(p)(k)  (7)

wherein u_(m)(k)′ is MFAC controller output, and Δu_(p)(k) isproportional controller output expressed by the following formulas:

$\begin{matrix}{{\Delta\;{u_{m}(k)}} = \frac{{{\rho\Phi}_{c}^{T}(k)}\left( {{y^{*}\left( {k + 1} \right)} - {y(k)}} \right)}{\lambda + {{\Phi_{c}(k)}}^{2}}} & (8) \\{{\Delta\;{u_{p}(k)}} = {{\beta\;{K\left( {{y^{*}\left( {k + 1} \right)} - {y(k)}} \right)}} - {\beta\;{K\left( {{y^{*}(k)} - {y\left( {k - 1} \right)}} \right)}}}} & (9)\end{matrix}$

proposing the following anti-windup algorithm as part of the proposedcontrol algorithm: stopping updating an integrator when an actuator isat an upper saturation limit and there is still a growing trend, or whenthe actuator is at a lower saturation limit and is still decreasing;otherwise, the integrator works normally; that is, in the case ofsaturation, only the integral operations that help to reduce the degreeof saturation are performed, and expressed by the following formulas:

$\begin{matrix}{{\Delta\;{u_{m}(k)}^{\prime}} = {\Delta\;{u_{m}(k)}{f(k)}}} & (10) \\{{f(k)} = \left\{ \begin{matrix}{0,} & {{{{u(k)} > {u_{-}\mspace{11mu}\max}} ⩓ {{\Delta\;{u(k)}} > 0}},{{{u(k)} < {u_{-}\mspace{11mu}\min}} ⩓ {{\Delta\;{u(k)}} < 0}}} \\{1,} & {otherwise}\end{matrix} \right.} & (11)\end{matrix}$

wherein u_max and u_min are the upper and lower limitations of theactuator;

proposing the following control solution based on formulas (6), (7), (8)and (9):

$\begin{matrix}{{{{\hat{\phi}}_{ii}(k)} = {{\hat{\phi}}_{ii}(1)}},{{{{if}\mspace{14mu}{{{\hat{\phi}}_{ii}(k)}}} < {b_{2}\mspace{14mu}{or}\mspace{14mu}{{{\hat{\phi}}_{ii}(k)}}} > {\alpha\; b_{2}\mspace{14mu}{or}\mspace{14mu}{sign}\mspace{14mu}\left( {{\hat{\phi}}_{ii}(k)} \right)} \neq {{sign}\mspace{14mu}\left( {{\hat{\phi}}_{ii}(1)} \right)\mspace{14mu} i}} = 1},\ldots\mspace{11mu},m} & (12) \\{{{{\hat{\phi}}_{ij}(k)} = {{\hat{\phi}}_{ij}(1)}},{{{if}\mspace{14mu}{{{\hat{\phi}}_{ij}(k)}}} > {b_{1}\mspace{14mu}{or}\mspace{14mu}{sign}\mspace{14mu}\left( {{\hat{\phi}}_{ij}(k)} \right)} \neq {{sign}\mspace{14mu}\left( {{\hat{\phi}}_{ij}(1)} \right)}},{i \neq j}} & (13) \\{{u(k)} = {{u\left( {k - 1} \right)} + {\frac{{{\rho\Phi}_{c}^{T}(k)}\left( {{y^{*}\left( {k + 1} \right)} - {y(k)}} \right)}{\lambda + {{\Phi_{c}(k)}}^{2}}{f(k)}} + {\beta\;{K\left( {{y^{*}\left( {k + 1} \right)} - {y(k)}} \right)}} - {\beta\;{K\left( {{y^{*}(k)} - {y\left( {k - 1} \right)}} \right)}}}} & (14)\end{matrix}$

wherein {circumflex over (ϕ)}_(ij)(1) is an initial value of {circumflexover (ϕ)}_(ij)(k), i=1, . . . , m; j=1, . . . , m;

step C: for the above improved control algorithm, analyzing theconvergence of tracking error and the stability of bounded input andbounded output through theoretical derivation;

firstly, defining the following output errors of the system:

e(k)=y*(k)−y(k)  (15)

substituting formula (2) and formula (14) into formula (15), and whenf(k)=1, obtaining:

$\begin{matrix}{{e\left( {k + 1} \right)} = {{{e(k)} - {{\Phi_{c}(k)}\Delta\;{u(k)}}} = {{\left\lbrack {I - \left( {\frac{{{\rho\Phi}_{c}(k)}{{\hat{\Phi}}_{c}^{T}(k)}}{\lambda + {{{\hat{\Phi}}_{c}(k)}}^{2}} + {{{\beta\Phi}_{c}(k)}K}} \right)} \right\rbrack{e(k)}} + {{{\beta\Phi}_{c}(k)}K\mspace{11mu}{e\left( {k - 1} \right)}}}}} & (16) \\\left. {D_{j} = {\left\{ {{z{{z -}}1} - \left( {\frac{\rho{\sum\limits_{i = 1}^{m}\;{{\phi_{ji}(k)}{{\hat{\phi}}_{ji}(k)}}}}{\lambda + {{\hat{\Phi}(k)}}^{2}} + {{{\beta\phi}_{jj}(k)}K_{jj}}} \right)} \right. \leq {\sum\limits_{{l = 1},{l \neq j}}^{m}\;{{\frac{\rho{\sum\limits_{i = 1}^{m}\;{{\phi_{ji}(k)}{{\hat{\phi}}_{li}(k)}}}}{\lambda + {{\hat{\Phi}(k)}}^{2}} + {\sum\limits_{i = 1}^{m}\;{{{\beta\phi}_{ji}(k)}K_{il}}}}}}}} \right\} & (17)\end{matrix}$

wherein z is a characteristic value of matrix I−(ρΦ_(c)(k){circumflexover (Φ)}_(c) ^(T)(k)/(λ+∥{circumflex over (Φ)}_(c)(k)∥²)+βΦ_(c)(k)K)and D_(j), j=1, 2, . . . , m is a Gershgorin disk;

formula (17) is equivalent to formula (18);

$\begin{matrix}\left. {D_{j}\left\{ {{z{z}} \leq {1 - {\left( {\frac{\rho{\sum\limits_{i = 1}^{m}\;{{\phi_{ji}(k)}{{\hat{\phi}}_{ji}(k)}}}}{\lambda + {{\hat{\Phi}(k)}}^{2}} + {{{\beta\phi}_{jj}(k)}K_{jj}}} \right){{{+ {\quad\quad}}\sum\limits_{{l = 1},{l \neq j}}^{m}}}\frac{\rho{\sum\limits_{i = 1}^{m}\;{{\phi_{ji}(k)}{{\hat{\phi}}_{li}(k)}}}}{\lambda + {{\overset{.}{\Phi}(k)}}^{2}}} + {\sum\limits_{i = 1}^{m}\;{{{\beta\phi}_{ji}(k)}K_{ii}}}}} \right.} \right\} & (18)\end{matrix}$

by resetting algorithms (12) and (13), obtaining |{circumflex over(ϕ)}_(ij)|≤b₁ and b₂≤|{circumflex over (ϕ)}_(ii)(k)|≤αb₂, i=1, . . . ,m;j=1, . . . ,m; i≠j; from assumption 1, obtaining|ϕ_(ij)|≤b₁,b₂≤|ϕ_(ii)(k)|≤αb₂, i=1, . . . ,m; j=1, . . . ,m; i≠j;

from the above conditions, obtaining the following inequalities

$\begin{matrix}{{1 - \left( {\frac{\rho{\sum\limits_{i = 1}^{m}\;{{{\phi_{ji}(k)}}{{{\hat{\phi}}_{ji}(k)}}}}}{\lambda + {{\hat{\Phi}(k)}}^{2}} + {{{\beta\phi}_{jj}(k)}K_{jj}}} \right)} \leq {1 - \left( {\frac{\rho{{\phi_{jj}(k)}}{{{\hat{\phi}}_{jj}(k)}}}{\lambda + {{\overset{.}{\Phi}(k)}}^{2}} + {{{\beta\phi}_{jj}(k)}K_{jj}}} \right)} \leq {1 - \left( {\frac{\rho\; b_{2}^{2}}{\lambda + {{\overset{.}{\Phi}(k)}}^{2}} + {\beta\; K_{\min}b_{2}}} \right)}} & (19)\end{matrix}$

by resetting algorithm formula (11) and assumption 1, obtaining{circumflex over (ϕ)}_(ji)(k)ϕ_(ji)(k)>0, i=1, . . . ,m; j=1, . . . ,m;therefore, there is a λ_(min), so that when λ>λ_(min), the followingequality holds:

${\frac{\rho{\sum_{i = 1}^{m}{{\phi_{ji}(k)}{{\hat{\phi}}_{ji}(k)}}}}{\lambda + {{\hat{\Phi}(k)}}^{2}} + {\beta{\phi_{jj}(k)}K_{jj}}} = {{\frac{\rho{\sum_{i = 1}^{m}{{❘{\phi_{ji}(k)}❘}{❘{{\hat{\phi}}_{ji}(k)}❘}}}}{\lambda + {{\hat{\Phi}(k)}}^{2}} + {\beta{\phi_{jj}(k)}K_{jj}}} \leq {{\rho\frac{{\alpha^{2}b_{2}^{2}} + {b_{1}^{2}\left( {m - 1} \right)}}{\lambda + {{\hat{\Phi}(k)}}^{2}}} + {\beta\alpha b_{2}K_{\max}}} < {{\rho\frac{{\alpha^{2}b_{2}^{2}} + {b_{1}^{2}\left( {m - 1} \right)}}{\lambda_{\min} + {{\hat{\Phi}(k)}}^{2}}} + {\beta\alpha b_{2}K_{\max}}} < 1}$

thus, selecting 0<ρ≤1 and λ>λ_(min) such that

$\begin{matrix}{{❘{1 - \left( {\frac{\rho{\sum_{i = 1}^{m}{{\phi_{ji}(k)}{{\hat{\phi}}_{ji}(k)}}}}{\lambda + {{\hat{\Phi}(k)}}^{2}} + {\beta{{\hat{\phi}}_{jj}(k)}K_{jj}}} \right)}❘} = {1 - \left( {\frac{\rho{\sum_{i = 1}^{m}{{\phi_{ji}(k)}{{\hat{\phi}}_{ji}(k)}}}}{\lambda + {{\hat{\Phi}(k)}}^{2}} + {\beta{{\hat{\phi}}_{jj}(k)}K_{jj}}} \right)}} & (23)\end{matrix}$

for any λ>λ_(min), the following inequalities hold obviously

$\begin{matrix}{0 < M_{1} \leq {\rho\frac{2\alpha{b_{1}^{2}\left( {m - 1} \right)}^{2}}{\lambda + {{\hat{\Phi}(k)}}^{2}}2\beta K_{\min}\alpha{b_{1}\left( {m - 1} \right)}} < {\frac{b_{2}^{2}}{\lambda + {{\hat{\Phi}(k)}}^{2}} + {2\beta K_{\min}\alpha{b_{1}\left( {m - 1} \right)}}} \leq {\frac{{\alpha^{2}b_{2}^{2}} + {b_{1}^{2}\left( {m - 1} \right)}}{\lambda + {{\hat{\Phi}(k)}}^{2}} + {2\beta K_{\min}\alpha{b_{1}\left( {m - 1} \right)}}} < {\frac{{\alpha^{2}b_{2}^{2}} + {b_{1}^{2}\left( {m - 1} \right)}}{\lambda_{\min} + {{\hat{\Phi}(k)}}^{2}} + {2\beta K_{\min}\alpha{b_{1}\left( {m - 1} \right)}}} < 1} & (24)\end{matrix}$

from formulas (21), (23) and (24), knowing

$\begin{matrix}{{{❘{1 - \left( {\frac{\rho{\sum_{i = 1}^{m}{{\phi_{ji}(k)}{{\hat{\phi}}_{ji}(k)}}}}{\lambda + {{\hat{\Phi}(k)}}^{2}} + {\beta{{\hat{\phi}}_{jj}(k)}K_{jj}}} \right)}❘} + {\sum_{{l = 1},{l \neq j}}^{m}{❘{\frac{\rho{\sum_{i = 1}^{m}{{\phi_{ji}(k)}{{\hat{\phi}}_{ji}(k)}x}}}{\lambda + {{\hat{\Phi}(k)}}^{2}} + {\beta{{\hat{\phi}}_{ji}(k)}K_{il}}}❘}}} < {1 - M_{1}} < 1} & (25)\end{matrix}$

from formulas (18) and (24), obtaining

$\begin{matrix}{{s\left\lbrack {I - \left( {\frac{\rho{\Phi_{c}(k)}{\Phi_{c}^{T}(k)}}{\lambda + {{{\hat{\Phi}}_{c}(k)}}^{2}} + {{{\beta\Phi}_{c}(k)}K}} \right)} \right\rbrack} \leq {1 - M_{1}}} & (26)\end{matrix}$

wherein s(M) is the spectral radius of matrix M;letting

$A = {{I - \left( {\frac{\rho{\Phi_{c}(k)}{\Phi_{c}^{T}(k)}}{\lambda + {{{\hat{\Phi}}_{c}(k)}}^{2}} + {\beta{\Phi_{c}(k)}K}} \right)}}_{p}$

and B=∥βΦ_(c)(k)K)∥_(v); from the conclusion of the spectral radius ofthe matrix, an any small positive number ε₁ exists, such that

$\begin{matrix}{A \leq {{s\left\lbrack {I - \left( {\frac{\rho{\Phi_{c}(k)}{\Phi_{c}^{T}(k)}}{\lambda + {{{\hat{\Phi}}_{c}(k)}}^{2}} + {{{\beta\Phi}_{c}(k)}K}} \right)} \right\rbrack} + \varepsilon_{1}} \leq {1 - M_{1} + \varepsilon_{1}} < 1} & (27)\end{matrix}$

wherein ∥M∥_(v) is the compatible norm of matrix M;

β exists such that B satisfies the following inequality:

1>1−A≤M ₁−ε₁ >B>0  (28)

from formulas (16) and (28), obtaining:

∥e(k+1)∥_(v) ≤A∥e(k)∥_(v) +B∥e(k−1)∥_(v)<(1−B)∥e(k)∥_(v)+B∥e(k−1)∥_(v)  (29)

after transposition, obtaining:

∥e(k+1)∥_(v) −∥e(k)∥_(v) <−B(∥e(k)∥_(v) −∥e(k−1)∥_(v))  (30)

based on formula (30), discussing the form of e(k) from the followingfour aspects:

1. when ∥e(k+1)∥_(v)>∥e(k)∥_(v) and ∥e(k)∥_(v)>∥e(k−1)∥_(v), obtaining

∥e(k+1)∥_(v) −∥e(k)∥_(v) >−B(∥e(k)∥_(v) −∥e(k−1)∥_(v))  (31)

which is the opposite of formula (30); therefore, this assumption doesnot exist;2. when ∥e(k+1)∥_(v)>∥e(k)∥_(v) and ∥e(k)∥_(v)<∥e(k−1)∥_(v) from formula(30), obtaining:

$\begin{matrix}{\frac{{{e\left( {k + 1} \right)}}_{v} - {{e(k)}}_{v}}{{{e\left( {k - 1} \right)}}_{v} - {{e(k)}}_{v}} < B < 1} & (32)\end{matrix}$

i.e., the decrease of e(k) is larger than the increase in three adjacentsampling points; and as a result, the overall trend is decreasing underthis situation;3. when ∥e(k+1)∥_(v)<∥e(k)∥_(v) and ∥e(k)∥_(v)<∥e(k−1)∥_(v), obtaining

$\begin{matrix}{\frac{{{e\left( {k + 1} \right)}}_{v}}{{{e(k)}}_{v}} < {1{and}\frac{{{e(k)}}_{v}}{{{e\left( {k - 1} \right)}}_{v}}} < 1} & (33)\end{matrix}$

which satisfies formula (30), and e(k) has a decreasing trend in thiscase;4. when ∥e(k+1)∥_(v)<∥e(k)∥_(v) and ∥e(k)∥_(v)>∥e(k−1)∥_(v), accordingto formula (30), this situation may exist; two possibilities exist inthe time of k+2 in detail: if ∥e(k+2)∥_(v)>∥e(k+1)∥_(v) exists,obtaining the same conclusion as the second case; if∥e(k+2)∥_(v)<∥e(k+1)∥_(v), obtaining the same conclusion as the thirdcase; in short, e(k) still has a decreasing trend in this case;

the above methods of proof are also applicable when f(k)=0; to sum up,the overall trend of error e(k) is decreasing; therefore, theconvergence of the error is proved;

step D: applying the above control algorithm to control of anaero-engine model, and selecting three different cases for resultcomparison to verify the effectiveness and superiority of the controlalgorithm; firstly, comparing the control effects of MFAC+Kp, CFDL-MFACand PID under the standard conditions to illustrate the effectiveness ofan improved controller; and then, comparing the control effects atdifferent heights and different delays to illustrate the superiority ofthe controller.

In the first case, the control effects of different algorithms arecompared under the standard conditions. The control effects of threealgorithms are shown in FIG. 2 under the nominal conditions of flightheight H=0, Ma=0, no noise and no delay. It can be seen that the risetime of MFAC+Kp algorithm is between MFAC and PID algorithm, but theadvantage over MFAC algorithm lies in smaller overshoot, which satisfiesthe strict stability requirement of the control algorithm for theperformance.

The second case is used to illustrate that the controller can adaptivelycontrol a broad flight envelope of the aircraft. The control effects areanalyzed at different flight heights. The results are shown in FIG. 3.From the simulation results, MFAC+Kp algorithm can realize stablecontrol for different flight heights. The higher the flight height, thegreater the overshoot, but the algorithm can still stabilize the systemoutput quickly with strong adaptive ability. In addition, compared withMFAC control effect under the same conditions, the control algorithm hasstronger stability.

The third case is used to verify the stable control for the model by thecontrol algorithm under the condition of delay. Four different delayvalues are selected to simulate under the flight conditions of H=10 andMa=1. The results show that MFAC+Kp algorithm can implement controlstably and quickly pointing at different degrees of delay. It can beseen from FIG. 4 that when the actuator is saturated, the proposedanti-windup algorithm can make the model get rid of a saturated regionquickly, but the MFAC algorithm takes a long time to get rid of thesaturated region under the same conditions, because of continuedoperation after saturation.

The present invention has the following beneficial effects:

(1) The CFDL-MFAC+Kp control algorithm improves the response speed androbustness of the original MFAC, and conducts theoretical analysis onthe basis of the existing MFAC proof of stability to prove the stabilityof the improved algorithm.

(2) Under the above control algorithm structure, the anti-windupalgorithm of the actuator is integrated and taken into account at thesame time, and the anti-windup effect of the above control algorithm isverified by experimental analysis.

DESCRIPTION OF DRAWINGS

FIG. 1 is a structural diagram of a controller.

FIG. 2 is an effect comparison diagram of three control algorithms ofMFAC+Kp, MFAC and PID.

FIG. 3 shows comparison of control effects at different flight heights.

FIG. 4 shows comparison of control effects at different delays.

DETAILED DESCRIPTION

To make the proposed technical solutions and the technical problemssolved by the present invention more clear, the technical solutions ofthe present invention are illustrated in detail below in combinationwith the drawings.

A structural block diagram of the improved control algorithm of thepresent invention is shown in FIG. 1. The controller mainly comprisesthree parts: MFAC, proportional control and anti-windup control. Thecontrol algorithm combines the advantages of three algorithms, canrealize stable and quick control even for a very complex nonlinear modeland has good robustness.

The specific composition of all parts of the control algorithm is asfollows:

(1) MFAC algorithm: at each sampling time point, the parameters of thecontrol algorithm are updated by the estimation algorithm, so that thecontrol algorithm can be changed adaptively to achieve a good controleffect on a control object, with certain robustness. However, due to theaddition of the estimation algorithm, the response time of thecontroller becomes slow and easily affected by disturbance. In order tosatisfy the requirements for the rapidity and the robustness of thecontroller, a proportional control link is considered to be added on thebasis of the control algorithm.

(2) Proportional control algorithm: this algorithm is simple inoperation and short in time consumption, reduces steady-state errors,accelerates control response, makes up for the deficiency of MFACalgorithm and improves the control performance.

(3) Anti-windup algorithm: due to the upper and lower limits of theactuator in the control system, the output of the control algorithm mayexceed the executive capacity of the actuator, making the actuator fallinto saturation, which will affect the response speed and controlaccuracy of the controller. The anti-windup algorithm can stop operationwhen the actuator reaches saturation, so that when the control algorithmprovides a normal instruction, the actuator can respond as quickly aswhen the actuator is not saturated.

The basic standard for measuring the control algorithm is the accuracy,stability and rapidity of control. The present invention also hasanti-windup performance while satisfying the above standard. Theimproved model-free adaptive control method of the present inventionmainly has the following advantages:

(1) Good accuracy. It can be seen from FIG. 3 and FIG. 4 that thecontrol algorithm in the present invention can achieve good controleffects at different heights and different delays, which indicates thatthe algorithm has good accuracy.

(2) Excellent stability. It can be seen from FIG. 2, FIG. 3 and FIG. 4that by comparing with MFAC and PID algorithms under the sameconditions, the control algorithm in the present invention has excellentstability, and can realize stable control at different flight heightsand different delays, and the stability is obviously better than theoriginal MFAC algorithm.

(3) Excellent rapidity. It can be seen from FIG. 3 and FIG. 4 that bycomparing with MFAC and PID algorithms under the same conditions, thecontrol algorithm in the present invention has excellent rapidity, andcan realize stable control at different flight heights and differentdelays, and the stability is obviously better than the original MFACalgorithm.

(4) Good anti-windup performance. It can be seen from FIG. 4 that theanti-windup algorithm in the present invention may stop accumulatingafter the actuator is saturated to prevent further saturation. After thecontroller outputs a normal value, the actuator can respond quickly, andthe response speed is higher than that of the original MFAC algorithm.

The improved model-free adaptive control method proposed in the presentinvention is provided below, which comprises the following specificsteps:

step A: analyzing the existing method for the compact dynamiclinearization model-free adaptive control, and from experimentalresults, finding that the application process has deficiencies inresponse time and stability;

expressing MIMO discrete-time nonlinear systems as follows:

y(k+1)=f(y(k), . . . ,y(k−n _(y)),u(k), . . . ,u(k−n _(u)))  (1)

wherein u(k) and y(k) are system inputs and system outputs at time k,respectively; n_(y) and n_(u) are two unknown integers; f( . . . )=(f₁(. . . ), . . . , f_(m)( . . . )) is an unknown nonlinear function;

when f has a continuous partial derivative condition and formula (1)satisfies a generalized Lipschitz condition, expressing formula (1) asthe following CFDL data model form:

$\begin{matrix}{{\Delta{y\left( {k + 1} \right)}} = {{\Phi_{c}(k)}\Delta{u(k)}{wherein}}} & (2)\end{matrix}$ ${{\Phi_{c}(k)} = {\begin{bmatrix}{\phi_{11}(k)} & {\phi_{12}(k)} & \ldots & {\phi_{1m}(k)} \\{\phi_{21}(k)} & {\phi_{22}(k)} & \ldots & {\phi_{2m}(k)} \\ \vdots & \vdots & \vdots & \vdots \\{\phi_{m1}(k)} & {\phi_{m2}(k)} & \ldots & {\phi_{mm}(k)}\end{bmatrix} \in R^{m \times m}}};$

firstly, proposing the following assumptions:

assumption 1: Φ_(c)(k) as a pseudo Jacobian matrix of the system shallbe a diagonal dominant matrix which satisfies the following conditions:|ϕ_(ij)|≤b₁,b₂≤|_(ii)(k)|≤αb₂,α≥1,b₂>b₁(2α+1)(m−1), i=1, . . . ,m, j=1,. . . ,m, i≠j; b₁ and b₂ are set as bounded constants, i and j are setas row and column indexes of the matrix; and the signs of all elementsin Φ_(c)(k) remain the same at any time k;

expressing a control input criterion function as formula (3):

J(u(k))=∥y*(k+1)−y(k+1)∥² +λ∥u(k)−u(k−1)∥²  (3)

wherein λ>0 represents a weight factor, which is used to punish thechange of excessive control input quantity; y*(k+1) is a desired outputsignal;

substituting formula (2) into formula (3), deriving u(k) and making theequation equal to zero to obtain a control input algorithm as follows:

$\begin{matrix}{{u(k)} = {{u\left( {k - 1} \right)} + \frac{{{\rho\Phi}_{c}^{T}(k)}\left( {{y^{*}\left( {k + 1} \right)} - {y(k)}} \right)}{\lambda + {{\Phi_{c}(k)}}^{2}}}} & (4)\end{matrix}$

considering the following parameter estimation criteria function:

J(Φ_(c)(k))=∥Δy(k)−Φ_(c)(k)Δu(k−1)∥²+μ∥Φ_(c)(k)−{circumflex over(Φ)}_(c)(k−1)∥²  (5)

wherein μ is a weight factor used to punish excessive changes in PJMestimates; {circumflex over (Φ)}_(c)(k) is an estimate of Φ_(c)(k);

deriving Φ_(c)(k) in formula (5) and making the equation equal to zeroto obtain a parameter estimation algorithm as follows:

$\begin{matrix}{{{\hat{\Phi}}_{c}(k)} = {{{\hat{\Phi}}_{c}\left( {k - 1} \right)} + \frac{\begin{matrix}{\eta\left( {{\Delta y(k)} - {{\hat{\Phi}}_{c}\left( {k - 1} \right)\Delta u\left( {k - 1} \right)}} \right)} \\{\Delta{u^{T}\left( {k - 1} \right)}}\end{matrix}}{\mu{{\Delta{u\left( {k - 1} \right)}}}^{2}}}} & (6)\end{matrix}$

conducting parameter estimation in each k by the above control parameterestimation algorithm to provide control inputs at the time; however, thecalculation of the parameter estimation algorithm needs to occupy acertain time, causing slow system response and causing the controlalgorithm to be limited in use for a system with a small requirement fora control period; and the system vibrates greatly under non-idealconditions from the experimental results;

step B: based on the above problems of slow response and vibration,considering the following improved solution;

Δu(k)=Δu _(m)(k)′+Δu _(p)(k)  (7)

wherein u_(m)(k)′ is MFAC controller output, and Δu_(p)(k) isproportional controller output expressed by the following formulas:

$\begin{matrix}{{\Delta{u_{m}(k)}} = \frac{{{\rho\Phi}_{c}^{T}(k)}\left( {{y^{*}\left( {k + 1} \right)} - {y(k)}} \right)}{\lambda + {{\Phi_{c}(k)}}^{2}}} & (8)\end{matrix}$ $\begin{matrix}{{\Delta{u_{p}(k)}} = {{\beta{K\left( {{y^{*}\left( {k + 1} \right)} - {y(k)}} \right)}} - {\beta{K\left( {{y^{*}(k)} - {y\left( {k - 1} \right)}} \right)}}}} & (9)\end{matrix}$

proposing the following control solution based on formulas (6) and (7):

$\begin{matrix}{{{{\hat{\phi}}_{ii}(k)} = {{\hat{\phi}}_{ii}(1)}},{{{if}{❘{{\hat{\phi}}_{ii}(k)}❘}} < {b_{2}{or}{❘{{\hat{\phi}}_{ii}(k)}❘}} > {\alpha b_{2}{or}}}} & (10)\end{matrix}$ sign(ϕ̂_(ii)(k)) ≠ sign(ϕ̂_(ii)(1))i = 1, …, m$\begin{matrix}{{{{\hat{\phi}}_{ij}(k)} = {{\hat{\phi}}_{ij}(1)}},{{{if}{}{❘{{\hat{\phi}}_{ij}(k)}❘}} > {b_{1}{or}{sign}\left( {{\hat{\phi}}_{ij}(k)} \right)} \neq {{sign}\left( {{\hat{\phi}}_{ij}(1)} \right)}},{i \neq j}} & (11)\end{matrix}$ $\begin{matrix}{{u(k)} = {{u\left( {k - 1} \right)} + {\frac{{{\rho\Phi}_{c}^{T}(k)}\left( {{y^{*}\left( {k + 1} \right)} - {y(k)}} \right)}{\lambda + {{\Phi_{c}(k)}}^{2}}{f(k)}} + {\beta{K\left( {{y^{*}\left( {k + 1} \right)} - {y(k)}} \right)}} - {\beta{K\left( {{y^{*}(k)} - {y\left( {k - 1} \right)}} \right)}}}} & (12)\end{matrix}$

wherein {circumflex over (ϕ)}_(ij)(1) is an initial value of {circumflexover (ϕ)}_(ij)(k), i=1, . . . , m; j=1, . . . , m;

proposing the following anti-windup algorithm as part of the proposedcontrol algorithm: stopping updating an integrator when an actuator isat an upper saturation limit and there is still a growing trend, or whenthe actuator is at a lower saturation limit and is still decreasing;otherwise, the integrator works normally; that is, in the case ofsaturation, only the integral operations that help to reduce the degreeof saturation are performed, and expressed by the following formulas:

$\begin{matrix}{{\Delta{u_{m}(k)}^{\prime}} = {\Delta{u_{m}(k)}{f(k)}}} & (13)\end{matrix}$ $\begin{matrix}{{f(k)} = \left\{ \begin{matrix}{0,} & \begin{matrix}{{{u(k)} > {{{u\_ max}\bigwedge\Delta}{u(k)}} > 0},} \\{{u(k)} < {{{u\_ min}\bigwedge\Delta}{u(k)}} < 0}\end{matrix} \\{1,} & {otherwise}\end{matrix} \right.} & (14)\end{matrix}$

wherein u_max and u_min are the upper and lower limits of the actuator;

step C: for the above improved control algorithm, analyzing theconvergence of tracking error and the stability of bounded input andbounded output through theoretical derivation;

firstly, defining the following output errors of the system:

e(k)=y*(k)−y(k)  (15)

substituting formula (2) and formula (14) into formula (15), and whenf(k)=1, obtaining:

$\begin{matrix}{\begin{matrix}{{e\left( {k + 1} \right)} = {{e(k)} - {{\Phi_{c}(k)}\Delta{u(k)}}}} \\{= \left\lbrack {I - \left( {\frac{{{\rho\Phi}_{c}(k)}{{\hat{\Phi}}_{c}^{T}(k)}}{\lambda + {{{\hat{\Phi}}_{c}(k)}}^{2}} + {\beta{\Phi_{c}(k)}K}} \right)} \right\rbrack} \\{{e(k)} + {\beta{\Phi_{c}(k)}K{e\left( {k - 1} \right)}}}\end{matrix}} & (16)\end{matrix}$ $\begin{matrix}{{D_{j} = \left\{ {{z{{z - {❘{1 - \left( {\frac{\rho{\sum_{i = 1}^{m}{{\phi_{ji}(k)}{{\hat{\phi}}_{ji}(k)}}}}{\lambda + {{\hat{\Phi}(k)}}^{2}} + {\beta{\phi_{jj}(k)}K_{jj}}} \right)}}}}} \leq {\sum\limits_{{l = 1},{l \neq j}}^{m}{❘{\frac{\rho{\sum_{i = 1}^{m}{{\phi_{ji}(k)}{{\hat{\phi}}_{li}(k)}}}}{\lambda + {{\hat{\Phi}(k)}}^{2}} + {\sum\limits_{i = 1}^{m}{\beta{\phi_{ji}(k)}K_{il}}}}❘}}} \right\}},} & (17)\end{matrix}$

wherein z is a characteristic value of matrix I−(ρΦ_(c)(k){circumflexover (Φ)}_(c) ^(T)(k)/(λ+∥{circumflex over (Φ)}_(c)(k)∥²)+βΦ_(c)(k)K)and D_(j), j=1, 2, . . . , m is a Gershgorin disk;

formula (17) is equivalent to formula (18);

$\begin{matrix}\left. {{{D_{j} = \left\{ {{z{❘❘}z{❘ \leq ❘}1} - \left( {\frac{\rho{\sum_{i = 1}^{m}{{\phi_{ji}(k)}{{\hat{\phi}}_{ji}(k)}}}}{\lambda + {{\hat{\Phi}(k)}}^{2}} + {\beta{\phi_{jj}(k)}K_{jj}}} \right)} \right.}} + {\sum\limits_{{l = 1},{l \neq j}}^{m}{❘{\frac{\rho{\sum_{i = 1}^{m}{{\phi_{ji}(k)}{{\hat{\phi}}_{li}(k)}}}}{\lambda + {{\hat{\Phi}(k)}}^{2}} + {\sum\limits_{i = 1}^{m}{\beta{\phi_{ji}(k)}K_{il}}}}❘}}} \right\} & (18)\end{matrix}$

by resetting algorithms (12) and (13), obtaining |{circumflex over(ϕ)}_(ij)|≤b₁ and b₂≤|{circumflex over (ϕ)}_(ii)(k)|≤αb₂, i=1, . . . ,m;j=1, . . . ,m; i≠j; from assumption 1, obtaining|ϕ_(ij)|≤b₁,b₂≤|ϕ_(ii)(k)|≤αb₂, i=1, . . . ,m; j=1, . . . ,m; i≠j;

from the above conditions, obtaining the following inequalities

$\begin{matrix}{{1 - \left( {\frac{\rho{\sum_{i = 1}^{m}{❘{{\phi_{ji}(k)}{\hat{❘{❘\phi}}}_{ji}(k)}❘}}}{\lambda + {{\hat{\Phi}(k)}}^{2}} + {\beta{\phi_{jj}(k)}K_{jj}}} \right)} \leq {1 - \left( {\frac{\rho{❘{\phi_{jj}(k)}❘}{❘{{\hat{\phi}}_{jj}(k)}❘}}{\lambda + {{\hat{\Phi}(k)}}^{2}} + {{{\beta\phi}_{jj}(k)}K_{jj}}} \right)} \leq {1 - \left( {\frac{\rho b_{2}^{2}}{\lambda + {{\hat{\Phi}(k)}}^{2}} + {\beta K_{\min}b_{2}}} \right)}} & (19)\end{matrix}$ $\begin{matrix}{{{\sum\limits_{{l = 1},{l \neq j}}^{m}{❘{\frac{\rho{\sum_{i = 1}^{m}{{\phi_{ji}(k)}{{\hat{\phi}}_{li}(k)}}}}{\lambda + {{\hat{\Phi}(k)}}^{2}} + {\sum\limits_{i = 1}^{m}{\beta{\phi_{ji}(k)}K_{il}}}}❘}} \leq {{\rho{\sum\limits_{{l = 1},{i \neq j}}^{m}\frac{\sum_{i = 1}^{m}{{❘{\phi_{ji}(k)}❘}{❘{{\hat{\phi}}_{li}(k)}❘}}}{\lambda + {{\hat{\Phi}(k)}}^{2}}}} + {\sum\limits_{{l = 1},{i \neq j}}^{m}{\beta{❘{\phi_{ji}(k)}❘}K_{li}}}}} = {{{\rho\frac{\sum_{{i = 1},{l \neq j}}^{m}{{❘{\phi_{jj}(k)}❘}{❘{{\hat{\phi}}_{lj}(k)}❘}}}{\lambda + {{\hat{\Phi}(k)}}^{2}}} + {\rho\frac{\sum_{{i = 1},{l \neq j}}^{m}{{❘{\phi_{ji}(k)}❘}{❘{{\hat{\phi}}_{il}(k)}❘}}}{\lambda + {{\hat{\Phi}(k)}}^{2}}} + {\rho\frac{\sum_{{l = 1},{l \neq j}}^{m}{\sum_{{i = 1},{i \neq j},l}^{m}{{❘{\phi_{ji}(k)}❘}{❘{{\hat{\phi}}_{li}(k)}❘}}}}{\lambda + {{\hat{\Phi}(k)}}^{2}}} + {\sum\limits_{{i = 1},{i \neq j}}^{m}{\beta{❘{\phi_{ji}(k)}❘}K_{ii}}}} \leq {{\rho\frac{\begin{matrix}{{2\alpha b_{1}{b_{2}\left( {m - 1} \right)}} +} \\{{b_{1}^{2}\left( {m - 1} \right)}\left( {m - 2} \right)}\end{matrix}}{\lambda + {{\hat{\Phi}(k)}}^{2}}} + {\beta K_{\max}{{b_{1}\left( {m - 1} \right)}.}}}}} & (20)\end{matrix}$ $\begin{matrix}{{{1 - \left( {\frac{\rho{\sum_{i = 1}^{m}{{❘{\phi_{ji}(k)}❘}{❘{{\hat{\phi}}_{ji}(k)}❘}}}}{\lambda + {{\hat{\Phi}(k)}}^{2}} + {\beta{\phi_{jj}(k)}K_{jj}}} \right) + {\sum_{{l = 1},{l \neq j}}^{m}{❘{\frac{\sum_{i = 1}^{m}{{\phi_{ji}(k)}{{\hat{\phi}}_{li}(k)}}}{\lambda + {{\hat{\Phi}(k)}}^{2}} + {\beta{\phi_{ji}(k)}K_{il}}}❘}}} \leq {1 - \begin{bmatrix}{{\rho\frac{b_{2}^{2} - {2\alpha b_{1}{b_{2}\left( {m - 1} \right)}} - {{b_{1}^{2}\left( {m - 1} \right)}\left( {m - 2} \right)}}{\lambda + {{\hat{\Phi}(k)}}^{2}}} +} \\{\beta K_{\min}\left( {b_{2} - {b_{1}\left( {m - 1} \right)}} \right)}\end{bmatrix}}} = {{{1 - \begin{bmatrix}{{\rho\frac{{b_{2}\left( {b_{2} - {2\alpha{b_{1}\left( {m - 1} \right)}}} \right)} - {{b_{1}^{2}\left( {m - 1} \right)}\left( {m - 2} \right)}}{\lambda + {{\hat{\Phi}(k)}}^{2}}} +} \\{\beta K_{\min}\left( {b_{2} - {b_{1}\left( {m - 1} \right)}} \right)}\end{bmatrix}} < {1 - \begin{bmatrix}{{\rho\frac{{b_{2}{b_{1}\left( {m - 1} \right)}} - {{b_{1}^{2}\left( {m - 1} \right)}\left( {m - 2} \right)}}{\lambda + {{\hat{\Phi}(k)}}^{2}}} +} \\{\beta K_{\min}\left( {b_{2} - {b_{1}\left( {m - 1} \right)}} \right)}\end{bmatrix}} < {1 - \begin{bmatrix}{{\rho\frac{{b_{2}{b_{1}\left( {m - 1} \right)}} - {{b_{1}^{2}\left( {m - 1} \right)}\left( {m - 1} \right)}}{\lambda + {{\hat{\Phi}(k)}}^{2}}} +} \\{\beta K_{\min}\left( {b_{2} - {b_{1}\left( {m - 1} \right)}} \right)}\end{bmatrix}}} = {{1 - \begin{bmatrix}{{\rho\frac{{b_{1}\left( {m - 1} \right)}\left( {b_{2} - {b_{1}\left( {m - 1} \right)}} \right.}{\lambda + {{\hat{\Phi}(k)}}^{2}}} +} \\{\beta K_{\min}\left( {b_{2} - {b_{1}\left( {m - 1} \right)}} \right)}\end{bmatrix}} < {1 - \left\lbrack {{\rho\frac{2\alpha{b_{1}^{2}\left( {m - 1} \right)}^{2}}{\lambda + {{\hat{\Phi}(k)}}^{2}}} + {2\beta K_{\min}\alpha{b_{1}\left( {m - 1} \right)}}} \right\rbrack}}}} & (21)\end{matrix}$

by resetting algorithm formula (11) and assumption 1, obtaining{circumflex over (ϕ)}_(ji)(k)ϕ_(ji)(k)>0, i=1, . . . ,m; j=1, . . . ,m;therefore, there is a λ_(min), so that when λ>λ_(min), the followingequality holds:

$\begin{matrix}{{\frac{\rho{\sum_{i = 1}^{m}{{\phi_{ji}(k)}{{\hat{\phi}}_{ji}(k)}}}}{\lambda + {{\hat{\Phi}(k)}}^{2}} + {\beta{\phi_{jj}(k)}K_{jj}}} = {{\frac{\rho{\sum_{i = 1}^{m}{❘{{\phi_{ji}(k)}{{\hat{❘{❘\phi}}}_{ji}(k)}}}}}{\lambda + {{\hat{\Phi}(k)}}^{2}} + {\beta{\phi_{jj}(k)}K_{jj}}} \leq {{\rho\frac{{\alpha^{2}b_{2}^{2}} + {b_{1}^{2}\left( {m - 1} \right)}}{\lambda + {{\hat{\Phi}(k)}}^{2}}} + {{\beta\alpha}b_{2}K_{\max}}} < {{\rho\frac{{\alpha^{2}b_{2}^{2}} + {b_{1}^{2}\left( {m - 1} \right)}}{\lambda_{\min} + {{\hat{\Phi}(k)}}^{2}}} + {\beta{\alpha b}_{2}K_{\max}}} < 1}} & (22)\end{matrix}$

thus, selecting 0<ρ≤1 and λ>λ_(min) such that

$\begin{matrix}{{❘{1 - \left( {\frac{\rho{\sum_{i = 1}^{m}{{\phi_{ji}(k)}{{\hat{\phi}}_{ji}(k)}}}}{\lambda + {{\hat{\Phi}(k)}}^{2}} + {{{\beta\phi}_{jj}(k)}K_{jj}}} \right)}❘} = {1 - \left( {\frac{\rho{\sum_{i = 1}^{m}{{\phi_{ji}(k)}{{\hat{\phi}}_{ji}(k)}}}}{\lambda + {{\hat{\Phi}(k)}}^{2}} + {{{\beta\phi}_{jj}(k)}K_{jj}}} \right)}} & (23)\end{matrix}$

for any λ>λ_(min), the following inequalities hold obviously

$\begin{matrix}{0 < M_{1} \leq {{\rho\frac{2\alpha{b_{1}^{2}\left( {m - 1} \right)}^{2}}{\lambda + {{\hat{\Phi}(k)}}^{2}}} + {2\beta K_{\min}\alpha{b_{1}\left( {m - 1} \right)}}} < {\frac{b_{2}^{2}}{\lambda + {{\hat{\Phi}(k)}}^{2}} + {2\beta K_{\min}\alpha{b_{1}\left( {m - 1} \right)}}} \leq {\frac{{\alpha^{2}b_{2}^{2}} + {b_{1}^{2}\left( {m - 1} \right)}}{\lambda + {{\hat{\Phi}(k)}}^{2}} + {2\beta K_{\min}\alpha{b_{1}\left( {m - 1} \right)}}} < {\frac{{\alpha^{2}b_{2}^{2}} + {b_{1}^{2}\left( {m - 1} \right)}}{\lambda_{\min} + {{\hat{\Phi}(k)}}^{2}} + {2\beta K_{\min}\alpha{b_{1}\left( {m - 1} \right)}}} < 1} & (24)\end{matrix}$

from formulas (21), (23) and (24), knowing

$\begin{matrix}{{{❘{1 - \left( {\frac{\rho{\sum_{i = 1}^{m}{{\phi_{ji}(k)}{{\hat{\phi}}_{ji}(k)}}}}{\lambda + {{\hat{\Phi}(k)}}^{2}} + {\beta{\phi_{jj}(k)}K_{jj}}} \right)}❘} + {\sum_{{l = 1},{l \neq j}}^{m}{❘{\frac{\rho{\sum_{i = 1}^{m}{{\phi_{ji}(k)}{{\hat{\phi}}_{ji}(k)}x}}}{\lambda + {{\hat{\Phi}(k)}}^{2}} + {\beta{{\hat{\phi}}_{ji}(k)}K_{il}}}❘}}} < {1 - M_{1}} < 1} & (25)\end{matrix}$

from formulas (18) and (24), obtaining

$\begin{matrix}{{s\left\lbrack {I - \left( {\frac{\rho{\Phi_{c}(k)}{{\hat{\Phi}}_{c}^{T}(k)}}{\lambda + {{\hat{\Phi}(k)}}^{2}} + {\beta{\Phi_{c}(k)}K}} \right)} \right\rbrack} \leq {1 - M_{1}}} & (26)\end{matrix}$

wherein s(M) is the spectral radius of matrix M;letting

$A = {{I - \left( {\frac{\rho{\Phi_{c}(k)}{{\hat{\Phi}}_{c}^{T}(k)}}{\lambda + {{\hat{\Phi}(k)}}^{2}} + {\beta{\Phi_{c}(k)}K}} \right.}}_{v}$

and B=∥βΦ_(c)(k)K)∥_(v); from the conclusion of the spectral radius ofthe matrix, an any small positive number ε₁ exists, such that

$\begin{matrix}{A \leq {{s\left\lbrack {I - \left( {\frac{\rho{\Phi_{c}(k)}{\Phi_{c}^{T}(k)}}{\lambda + {{\hat{\Phi}(k)}}^{2}} + {\beta{\Phi_{c}(k)}K}} \right)} \right\rbrack} + \varepsilon_{1}} \leq {1 - M_{1} + \varepsilon_{1}} < 1} & (27)\end{matrix}$

wherein ∥M∥_(v) is the compatible norm of matrix M;

β exists such that B satisfies the following inequality:

1>1−A≤M ₁−ε₁ >B>0  (28)

from formulas (16) and (28), obtaining:

∥e(k+1)∥_(v) ≤A∥e(k)∥_(v) +B∥e(k−1)∥_(v)<(1−B)∥e(k)∥_(v)+B∥e(k−1)∥_(v)  (29)

after transposition, obtaining:

∥e(k+1)∥_(v) −∥e(k)∥_(v) <−B(∥e(k)∥_(v) −∥e(k−1)∥_(v))  (30)

based on formula (30), discussing the form of e(k) from the followingfour aspects:

1. when ∥e(k+1)∥_(v)>∥e(k)∥_(v) and ∥e(k)∥_(v)>∥e(k−1)∥_(v), obtaining

∥e(k+1)∥_(v) −∥e(k)∥_(v) >−B(∥e(k)∥_(v) −∥e(k−1)∥_(v))  (31)

which is the opposite of formula (30); therefore, this assumption doesnot exist;2. when ∥e(k+1)∥_(v)>∥e(k)∥_(v) and ∥e(k)∥_(v)<∥e(k−1)∥_(v) from formula(30), obtaining:

$\begin{matrix}{\frac{{{e\left( {k + 1} \right)}}_{v} - {{e(k)}}_{v}}{{{e\left( {k - 1} \right)}}_{v} - {{e(k)}}_{v}} < B < 1} & (32)\end{matrix}$

i.e., the decrease of e(k) is larger than the increase in three adjacentsampling points; and as a result, the overall trend is decreasing underthis situation;3. when ∥e(k+1)∥_(v)<∥e(k)∥_(v) and ∥e(k)∥_(v)<∥e(k−1)∥_(v), obtaining

$\begin{matrix}{\frac{{{e\left( {k + 1} \right)}}_{v}}{{{e(k)}}_{v}} < {1{and}\frac{{{e(k)}}_{v}}{{{e\left( {k - 1} \right)}}_{v}}} < 1} & (33)\end{matrix}$

which satisfies formula (30), and e(k) has a decreasing trend in thiscase;4. when ∥e(k+1)∥_(v)<∥e(k)∥_(v) and ∥e(k)∥_(v)>∥e(k−1)∥_(v), accordingto formula (30), this situation may exist; two possibilities exist inthe time of k+2 in detail: if ∥e(k+2)∥_(v)>∥e(k+1)∥_(v) exists,obtaining the same conclusion as the second case; if∥e(k+2)∥_(v)<∥e(k+1)∥_(v), obtaining the same conclusion as the thirdcase; in short, e(k) still has a decreasing trend in this case;

the above methods of proof are also applicable when f(k)=0; to sum up,the overall trend of error e(k) is decreasing; therefore, theconvergence of the error is proved;

step D: applying the above control algorithm to control of anaero-engine model, and selecting three different cases for resultcomparison to verify the effectiveness and superiority of the controlalgorithm; firstly, comparing the control effects of MFAC+Kp, CFDL-MFACand PID under the standard conditions to illustrate the effectiveness ofan improved controller; and then, comparing the control effects atdifferent heights and different delays to illustrate the superiority ofthe controller.

In the first case, the control effects of different algorithms arecompared under the standard conditions. The control effects of threealgorithms are shown in FIG. 2 under the nominal conditions of flightheight H=0, Ma=0, no noise and no delay. It can be seen that the risetime of MFAC+Kp algorithm is between MFAC and PID algorithm, but theadvantage over MFAC algorithm lies in smaller overshoot, which satisfiesthe strict stability requirement of the control algorithm for theperformance.

The second case is used to illustrate that the controller can adaptivelycontrol a broad flight envelope of the aircraft. The control effects areanalyzed at different flight heights. The results are shown in FIG. 3.From the simulation results, MFAC+Kp algorithm can realize stablecontrol for different flight heights. The higher the flight height, thegreater the overshoot, but the algorithm can still stabilize the systemoutput quickly with strong adaptive ability. In addition, compared withMFAC control effect under the same conditions, the control algorithm hasstronger stability.

The third case is used to verify the stable control for the model by thecontrol algorithm under the condition of delay. Four different delayvalues are selected to simulate under the flight conditions of H=10 andMa=1. The results show that MFAC+Kp algorithm can implement controlstably and quickly pointing at different degrees of delay. It can beseen from FIG. 4 that when the actuator is saturated, the proposedanti-windup algorithm can make the model get rid of a saturated regionquickly, but the MFAC algorithm takes a long time to get rid of thesaturated region under the same conditions, because of continuedoperation after saturation.

In conclusion, the improved model-free adaptive control method of thepresent invention proposes a new model-free adaptive control methodwhich improves the overshoot oscillation of MFAC by adding proportionalcontrol. At the same time, the present invention integrates the idea ofintegral anti-windup to improve the control performance. It is provedthrough strict analysis that the improved control algorithm has trackingerror convergence and BIBO stability under the condition of satisfyingthe assumption. Finally, the improved MFAC is applied to the control ofthe aero-engine model. Three experiments are carried out from differentperspectives to verify the anti-windup performance, rapidity andstability of the control algorithm of the present invention at differentflight heights and different delays. The results are superior to theMFAC algorithm and the PID algorithm. The results show that the controlalgorithm proposed herein has stable and quick control effects on theaero-engine control system and the effectiveness of the algorithm isverified.

It should be noted that those skilled in the art should understand thatthe above embodiments are only used for illustrating the technicalsolutions of the present invention, rather than limiting the presentinvention. Different technical features that appear in differentembodiments can be combined to obtain beneficial effects. On the basisof the description and the claims, the researchers in the art shallunderstand and realize other varied embodiments of disclosed embodimentsin combination with the drawings. It should be noted that the presentinvention is described in detail by referring to the above embodiments,and the amendments to the technical solution mentioned in each of theabove embodiments or the equivalent replacements for part of or all thetechnical features therein do not enable the essence of thecorresponding technical solution to depart from the scope of thetechnical solution of various embodiments of the present invention.

1. An improved model-free adaptive control method, comprising steps of:step A: analyzing the existing method for the compact dynamiclinearization model-free adaptive control, and from experimentalresults, finding that the application process has deficiencies inresponse time and stability; expressing MIMO discrete-time nonlinearsystems as follows:y(k+1)=f(y(k), . . . ,y(k−n _(y)),u(k), . . . ,u(k−n _(u)))  (1) whereinu(k) and y(k) are system inputs and system outputs at time k,respectively; n_(y) and n_(u) are two unknown integers; f( . . . )=(f₁(. . . ), . . . , f_(m)( . . . )) is an unknown nonlinear function; whenf has a continuous partial derivative condition and formula (1)satisfies a generalized Lipschitz condition, expressing formula (1) asthe following CFDL data model form: $\begin{matrix}{{\Delta{y\left( {k + 1} \right)}} = {{\Phi_{c}(k)}\Delta{u(k)}{wherein}}} & (2)\end{matrix}$ ${{\Phi_{c}(k)} = {\begin{bmatrix}{\phi_{11}(k)} & {\phi_{12}(k)} & \ldots & {\phi_{1m}(k)} \\{\phi_{21}(k)} & {\phi_{22}(k)} & \ldots & {\phi_{2m}(k)} \\ \vdots & \vdots & \vdots & \vdots \\{\phi_{m1}(k)} & {\phi_{m2}(k)} & \ldots & {\phi_{mm}(k)}\end{bmatrix} \in R^{m \times m}}};$ firstly, proposing the followingassumptions: assumption 1: Φ_(c)(k) as a pseudo Jacobian matrix of thesystem shall be a diagonal dominant matrix which satisfies the followingconditions: |ϕ_(ij)|≤b₁,b₂≤|_(ii)(k)|≤αb₂,α≥1,b₂>b₁(2α+1)(m−1), i=1, . .. , m, j=1, . . . , m, i≠j; b₁ and b₂ are set as bounded constants, iand j are set as row and column indexes of the matrix; and the signs ofall elements in Φ_(c)(k) remain the same at any time k; expressing acontrol input criterion function as formula (3):J(u(k))=∥y*(k+1)−y(k+1)∥² +λ∥u(k)−u(k−1)∥²  (3) wherein λ>0 represents aweight factor, which is used to punish the change of excessive controlinput quantity; y*(k+1) is a desired output signal; substituting formula(2) into formula (3), deriving u(k) and making the equation equal tozero to obtain a control input algorithm as follows: $\begin{matrix}{{u(k)} = {{u\left( {k - 1} \right)} + \frac{{{\rho\Phi}_{c}^{T}(k)}\left( {{y^{*}\left( {k + 1} \right)} - {y(k)}} \right)}{\lambda + {{\Phi_{c}(k)}}^{2}}}} & (4)\end{matrix}$ considering the following parameter estimation criteriafunction:J(Φ_(c)(k))=∥Δy(k)−Φ_(c)(k)Δu(k−1)∥²+μ∥Φ_(c)(k)−{circumflex over(Φ)}_(c)(k−1)∥²  (5) wherein μ is a weight factor used to punishexcessive changes in PJM estimates; {circumflex over (Φ)}_(c)(k) is anestimate of Φ_(c)(k); deriving Φ_(c)(k) in formula (5) and making theequation equal to zero to obtain a parameter estimation algorithm asfollows: $\begin{matrix}{{{\hat{\Phi}}_{c}(k)} = {{{\hat{\Phi}}_{c}\left( {k - 1} \right)} + \frac{{\eta\left( {{\Delta{y(k)}} - {{{\hat{\Phi}}_{c}\left( {k - 1} \right)}\Delta{u\left( {k - 1} \right)}}} \right)}\Delta{u^{T}\left( {k - 1} \right)}}{\mu + {{\Delta{u\left( {k - 1} \right)}}}^{2}}}} & (6)\end{matrix}$ conducting parameter estimation in each k by the abovecontrol parameter estimation algorithm to provide control inputs at thetime; however, the calculation of the parameter estimation algorithmneeds to occupy a certain time, causing slow system response and causingthe control algorithm to be limited in use for a system with a smallrequirement for a control period; and the system vibrates greatly undernon-ideal conditions from the experimental results; step B: based on theabove problems of slow response and vibration, considering the followingimproved solution;Δu(k)=Δu _(m)(k)′+Δu _(p)(k)  (7) wherein u_(m)(k)′ is MFAC controlleroutput, and Δu_(p)(k) is proportional controller output expressed by thefollowing formulas: $\begin{matrix}{{\Delta{u_{m}(k)}} = \frac{{{\rho\Phi}_{c}^{T}(k)}\left( {{y^{*}\left( {k + 1} \right)} - {y(k)}} \right)}{\lambda + {{\Phi_{c}(k)}}^{2}}} & (8)\end{matrix}$ $\begin{matrix}{{\Delta{u_{p}(k)}} = {{\beta{K\left( {{y^{*}\left( {k + 1} \right)} - {y(k)}} \right)}} - {\beta{K\left( {{y^{*}(k)} - {y\left( {k - 1} \right)}} \right)}}}} & (9)\end{matrix}$ proposing the following anti-windup algorithm as part ofthe proposed control algorithm: stopping updating an integrator when anactuator is at an upper saturation limit and there is still a growingtrend, or when the actuator is at a lower saturation limit and is stilldecreasing; otherwise, the integrator works normally; that is, in thecase of saturation, only the integral operations that help to reduce thedegree of saturation are performed, and expressed by the followingformulas: $\begin{matrix}{{\Delta{u_{m}(k)}^{\prime}} = {\Delta{u_{m}(k)}{f(k)}}} & (10)\end{matrix}$ $\begin{matrix}{{f(k)} = \left\{ \begin{matrix}{0,} & \begin{matrix}{{{u(k)} > {{{u\_ max}\bigwedge\Delta}{u(k)}} > 0},} \\{{u(k)} < {{{u\_ min}\bigwedge\Delta}{u(k)}} < 0}\end{matrix} \\{1,} & {otherwise}\end{matrix} \right.} & (11)\end{matrix}$ wherein u_max and u_min are the upper and lowerlimitations of the actuator; proposing the following control solutionbased on formulas (6), (7), (8) and (9): $\begin{matrix}{{{{{{{{\hat{\phi}}_{ii}(k)} = {{\hat{\phi}}_{ii}(1)}},{{{if}{❘{{\hat{\phi}}_{ii}(k)}❘}} < {b_{2}{or}}}}❘}{{\hat{\phi}}_{ii}(k)}}❘} > {\alpha b_{2}{or}}} & (12)\end{matrix}$ sign(ϕ̂_(ii)(k)) ≠ sign(ϕ̂_(ii)(1))i = 1, …, m$\begin{matrix}{{{{\hat{\phi}}_{ij}(k)} = {{\hat{\phi}}_{ij}(1)}},{{{if}{❘{{\hat{\phi}}_{ij}(k)}❘}} > {b_{1}{or}{sign}\left( {{\hat{\phi}}_{ij}(k)} \right)} \neq {{sign}\left( {{\hat{\phi}}_{ij}(1)} \right)}},{i \neq j}} & (13)\end{matrix}$${u(k)} = {{u\left( {k - 1} \right)} + {\frac{{{\rho\Phi}_{c}^{T}(k)}\left( {{y^{*}\left( {k + 1} \right)} - {y(k)}} \right)}{\lambda + {{\Phi_{c}(k)}}^{2}}{f(k)}} + {\beta{K\left( {{y^{*}\left( {k + 1} \right)} - {y(k)}} \right)}} - {\beta{K\left( {{y^{*}(k)} - {y\left( {k - 1} \right)}} \right)}}}$wherein {circumflex over (ϕ)}_(ij)(1) is an initial value of {circumflexover (ϕ)}_(ij)(k), i=1, . . . , m; j=1, . . . , m; step C: for the aboveimproved control algorithm, analyzing the convergence of tracking errorand the stability of bounded input and bounded output throughtheoretical derivation; firstly, defining the following output errors ofthe system:e(k)=y*(k)−y(k)  (15) substituting formula (2) and formula (14) intoformula (15), and when f(k)=1, obtaining: $\begin{matrix}{\begin{matrix}{{e\left( {k + 1} \right)} = {{e(k)} - {{\Phi_{e}(k)}\Delta{u(k)}}}} \\{= \left\lbrack {I - \left( {\frac{{{\rho\Phi}_{e}(k)}{{\hat{\Phi}}_{c}^{T}(k)}}{\lambda + {{{\hat{\Phi}}_{c}(k)}}^{2}} + {\beta{\Phi_{c}(k)}K}} \right)} \right.} \\{{e(k)} + {{{\beta\Phi}_{c}(k)}{{Ke}\left( {k - 1} \right)}}}\end{matrix}} & (16)\end{matrix}$ $\begin{matrix}{D_{j} = \left\{ {{z{\left. {z -} \middle| {1 - \left( {\frac{\rho{\sum_{i = 1}^{m}{{\phi_{ji}(k)}{{\hat{\phi}}_{ji}(k)}}}}{\lambda + {{\Phi(k)}}^{2}} + {\beta{\phi_{jj}(k)}K_{jj}}} \right)} \right.}} \leq {\overset{m}{\sum\limits_{{l = 1},{i \neq j}}}{❘{\frac{\rho{\sum_{i = 1}^{m}{{\phi_{ji}(k)}{{\hat{\phi}}_{li}(k)}}}}{\lambda + {{\hat{\Phi}(k)}}^{2}} + {\sum\limits_{i = 1}^{m}{\beta{\phi_{ji}(k)}K_{il}}}}❘}}} \right\}} & (17)\end{matrix}$ wherein z is a characteristic value of matrixI−(ρΦ_(c)(k){circumflex over (Φ)}_(c) ^(T)(k)/(λ+∥{circumflex over(Φ)}_(c)(k)∥²)+βΦ_(c)(k)K) and D_(j), j=1, 2, . . . , m is a Gershgorindisk; formula (17) is equivalent to formula (18); $\begin{matrix}\left. {{{{{{{D_{i} = \left\{ {z{{{z{❘ \leq ❘}1} - \left( {\frac{\rho{\sum_{i = 1}^{m}{{\phi_{ji}(k)}{{\hat{\phi}}_{ji}(k)}}}}{\lambda + {{\hat{\Phi}(k)}}^{2}} + {\beta{\phi_{jj}(k)}K_{jj}}} \right)}}} \right.}❘} + \overset{m}{\sum\limits_{{l = 1},{i \neq j}}}}❘}\frac{\rho{\sum_{i = 1}^{m}{{\phi_{ji}(k)}{{\hat{\phi}}_{li}(k)}}}}{\lambda + {{\hat{\Phi}(k)}}^{2}}} + {\sum\limits_{i = 1}^{m}{\beta{\phi_{ji}(k)}K_{il}}}}❘} \right\} & (18)\end{matrix}$ by resetting algorithms (12) and (13), obtaining|{circumflex over (ϕ)}_(ij)|≤b₁ and b₂≤|{circumflex over(ϕ)}_(ii)(k)|≤αb₂, i=1, . . . ,m; j=1, . . . ,m; i≠j; from assumption 1,obtaining |ϕ_(ij)|≤b₁,b₂≤|ϕ_(ii)(k)|≤αb₂, i=1, . . . ,m; j=1, . . . ,m;i≠j; from the above conditions, obtaining the following inequalities$\begin{matrix}{{1 - \left( {\frac{\rho{\sum_{i = 1}^{m}{{❘{\phi_{ji}(k)}❘}{❘{{\hat{\phi}}_{ji}(k)}❘}}}}{\lambda + {{\hat{\Phi}(k)}}^{2}} + {\beta{\phi_{jj}(k)}K_{jj}}} \right)} \leq {1 - \left( {\frac{\rho{❘{\phi_{jj}(k)}❘}{❘{{\hat{\phi}}_{jj}(k)}❘}}{\lambda + {{\hat{\Phi}(k)}}^{2}} + {\beta{\phi_{jj}(k)}K_{jj}}} \right)} \leq {1 - \left( {\frac{\rho b_{2}^{2}}{\lambda + {{\hat{\Phi}(k)}}^{2}} + {\beta K_{\min}b_{2}}} \right)}} & (19)\end{matrix}$ $\begin{matrix}\begin{matrix}{{{\overset{m}{\sum\limits_{{l = 1},{i \neq j}}}{❘{\frac{{\rho{\sum_{i = 1}^{m}{{\phi_{ji}(k)}{{\hat{\phi}}_{li}(k)}}}}❘}{\lambda + {{\hat{\Phi}(k)}}^{2}} + {\sum\limits_{i = 1}^{m}{\beta{\phi_{ji}(k)}K_{il}}}}❘}} \leq {{\rho{\overset{m}{\sum\limits_{{l = 1},{i \neq j}}}\frac{\sum_{i = 1}^{m}{{❘{\phi_{ji}(k)}❘}{❘{{\hat{\phi}}_{li}(k)}❘}}}{\lambda + {{\hat{\Phi}(k)}}^{2}}}} + {\overset{m}{\sum\limits_{{l = 1},{i \neq j}}}{\beta{❘{\phi_{jl}(k)}❘}K_{il}}}}} = {{{\rho\frac{\sum_{{i = 1},{l \neq j}}^{m}{{❘{\phi_{ji}(k)}❘}{❘{{\hat{\phi}}_{lj}(k)}❘}}}{\lambda + {{\hat{\Phi}(k)}}^{2}}} + {\rho\frac{\sum_{{i = 1},{l \neq j}}^{m}{{❘{\phi_{jl}(k)}❘}{❘{{\hat{\phi}}_{li}(k)}❘}}}{\lambda + {{\hat{\Phi}(k)}}^{2}}} + {\rho\frac{\begin{matrix}{\sum_{{l = 1},{l \neq j}}^{m}\sum_{{i = 1},{i \neq j},l}^{m}} \\{{❘{\phi_{ji}(k)}❘}{❘{{\hat{\phi}}_{li}(k)}❘}}\end{matrix}}{\lambda + {{\hat{\Phi}(k)}}^{2}}} + {\overset{m}{\sum\limits_{{l = 1},{l \neq j}}}{\beta{❘{\phi_{jl}(k)}❘}K_{il}}}} \leq {{\rho\frac{\begin{matrix}{{2\alpha b_{1}{b_{2}\left( {m - 1} \right)}} +} \\{{b_{1}^{2}\left( {m - 1} \right)}\left( {m - 2} \right)}\end{matrix}}{\lambda + {{\hat{\Phi}(k)}}^{2}}} + {\beta K_{\max}{{b_{1}\left( {m - 1} \right)}.}}}}} & \end{matrix} & (20)\end{matrix}$ $\begin{matrix}{{{1 - \left( {\frac{\rho{\sum_{i = 1}^{m}{{❘{\phi_{ji}(k)}❘}{❘{{\hat{\phi}}_{ji}(k)}❘}}}}{\lambda + {{\hat{\Phi}(k)}}^{2}} + {\beta{\phi_{jj}(k)}K_{jj}}} \right) + {\sum_{{l = 1},{l \neq j}}^{m}{❘{\frac{\sum_{i = 1}^{m}{{\phi_{ji}(k)}{{\hat{\phi}}_{li}(k)}}}{\lambda + {{\hat{\Phi}(k)}}^{2}} + {{{\beta\phi}_{ji}(k)}K_{il}}}❘}}} \leq {1 - \left\lbrack \begin{matrix}{{\rho\frac{b_{2}^{2} - {2\alpha b_{1}{b_{2}\left( {m - 1} \right)}} - {{b_{1}^{2}\left( {m - 1} \right)}\left( {m - 2} \right)}}{\lambda + {{\hat{\Phi}(k)}}^{2}}} +} \\{\beta K_{\min}\left( {b_{2} - {b_{1}\left( {m - 1} \right)}} \right)}\end{matrix} \right\rbrack}} = {{{1 - \begin{bmatrix}{{\rho\frac{{b_{2}\left( {b_{2} - {2\alpha{b_{1}\left( {m - 1} \right)}}} \right)} - {{b_{1}^{2}\left( {m - 1} \right)}\left( {m - 2} \right)}}{\lambda + {{\hat{\Phi}(k)}}^{2}}} +} \\{\beta K_{\min}\left( {b_{2} - {b_{1}\left( {m - 1} \right)}} \right)}\end{bmatrix}} < {1 - \begin{bmatrix}{{\rho\frac{{b_{2}{b_{1}\left( {m - 1} \right)}} - {{b_{1}^{2}\left( {m - 1} \right)}\left( {m - 2} \right)}}{\lambda + {{\hat{\Phi}(k)}}^{2}}} +} \\{\beta K_{\min}\left( {b_{2} - {b_{1}\left( {m - 1} \right)}} \right)}\end{bmatrix}} < {1 - \begin{bmatrix}{{\rho\frac{{b_{2}{b_{1}\left( {m - 1} \right)}} - {{b_{1}^{2}\left( {m - 1} \right)}\left( {m - 2} \right)}}{\lambda + {{\hat{\Phi}(k)}}^{2}}} +} \\{\beta K_{\min}\left( {b_{2} - {b_{1}\left( {m - 1} \right)}} \right)}\end{bmatrix}}} = {{1 - \left\lbrack {{\rho\frac{{b_{1}\left( {m - 1} \right)}\left( {b_{2} - {b_{1}\left( {m - 1} \right)}} \right)}{\lambda + {{\hat{\Phi}(k)}}^{2}}} + {\beta{K_{\min}\left( {b_{2} - {b_{1}\left( {m - 1} \right)}} \right)}}} \right\rbrack} < {1 - \left\lbrack {{\rho\frac{2\alpha{b_{1}^{2}\left( {m - 1} \right)}^{2}}{\lambda + {{\hat{\Phi}(k)}}^{2}}} + {2\beta K_{\min}\alpha{b_{1}\left( {m - 1} \right)}}} \right\rbrack}}}} & (21)\end{matrix}$ from resetting algorithm formula (11) and assumption 1,obtaining {circumflex over (ϕ)}_(ji)(k)ϕ_(ji)(k)>0, i=1, . . . , m; j=1,. . . , m; therefore, there is a λ_(min), so that when λ>λ_(min), thefollowing equality holds: $\begin{matrix}{{\frac{\rho{\sum_{i = 1}^{m}{{\phi_{ji}(k)}{{\hat{\phi}}_{ji}(k)}}}}{\lambda + {{\hat{\Phi}(k)}}^{2}} + {\beta{\phi_{jj}(k)}K_{lj}}} = {\frac{\rho{\sum_{i = 1}^{m}{{❘{\phi_{ji}(k)}❘}{❘{{\hat{\phi}}_{ji}(k)}❘}}}}{\lambda + {{\hat{\Phi}(k)}}^{2}} = {{\beta{\phi_{jj}(k)}K_{ji}} \leq {{\rho\frac{{\alpha^{2}b_{2}^{2}} + {b_{1}^{2}\left( {m - 1} \right)}}{\lambda + {{\hat{\Phi}(k)}}^{2}}} + {\beta\alpha b_{2}K_{\max}}} < {{\rho\frac{{\alpha^{2}b_{2}^{2}} + {b_{1}^{2}\left( {m - 1} \right)}}{\lambda_{\min} + {{\hat{\Phi}(k)}}^{2}}} + {\beta\alpha b_{2}K_{\max}}} < 1}}} & (22)\end{matrix}$ thus, selecting 0<ρ≤1 and λ>λ_(min) such that$\begin{matrix}{{❘{1 - \left( {\frac{\rho{\sum_{i = 1}^{m}{{\phi_{ji}(k)}{{\hat{\phi}}_{ji}(k)}}}}{\lambda + {{\hat{\Phi}(k)}}^{2}} + {\beta{\phi_{jj}(k)}K_{lj}}} \right)}❘} = {1 - \left( {\frac{\rho{\sum_{i = 1}^{m}{{\phi_{ji}(k)}{{\hat{\phi}}_{ji}(k)}}}}{\lambda + {{\hat{\Phi}(k)}}^{2}} + {\beta{\phi_{jj}(k)}K_{jj}}} \right)}} & (23)\end{matrix}$ for any λ>λ_(min), the following inequalities holdobviously $\begin{matrix}{0 < M_{1} \leq {{\rho\frac{2\alpha{b_{1}^{2}\left( {m - 1} \right)}^{2}}{\lambda + {{\hat{\Phi}(k)}}^{2}}} + {2\beta K_{\min}\alpha{b_{1}\left( {m - 1} \right)}}} < {\frac{b_{2}^{2}}{\lambda + {{\hat{\Phi}(k)}}^{2}} + {2\beta K_{\min}\alpha{b_{1}\left( {m - 1} \right)}}} \leq {\frac{{\alpha^{2}b_{2}^{2}} + {b_{1}^{2}\left( {m - 1} \right)}}{\lambda + {{\hat{\Phi}(k)}}^{2}} + {2\beta K_{\min}\alpha{b_{1}\left( {m - 1} \right)}}} < {\frac{{\alpha^{2}b_{2}^{2}} + {b_{1}^{2}\left( {m - 1} \right)}}{\lambda_{\min} + {{\hat{\Phi}(k)}}^{2}} + {2\beta K_{\min}\alpha{b_{1}\left( {m - 1} \right)}}} < 1} & (24)\end{matrix}$ from formulas (21), (23) and (24), knowing $\begin{matrix}{{{{{{{❘{1 - {\begin{pmatrix}{\frac{\rho{\sum_{i = 1}^{m}{{\phi_{ji}(k)}{{\hat{\phi}}_{ji}(k)}}}}{\lambda + {{\hat{\Phi}(k)}}^{2}} +} \\{\beta\phi_{jj}(k)K_{jj}}\end{pmatrix}}}❘} + {{\sum_{{l = 1},{l \neq j}}^{m}}}}❘}\frac{\rho{\sum_{i = 1}^{m}{{\phi_{ji}(k)}{{\hat{\phi}}_{li}(k)}}}}{\lambda + {{\hat{\Phi}(k)}}^{2}}} + {\beta{\phi_{ji}(k)}K_{il}}}❘} < {1 - M_{1}} < 1} & (25)\end{matrix}$ from formulas (18) and (24), obtaining $\begin{matrix}{{s\left\lbrack {I - \left( {\frac{\rho{\Phi_{c}(k)}{{\hat{\Phi}}_{c}^{T}(k)}}{\lambda + {{\hat{\Phi}(k)}}^{2}} + {\beta{\Phi_{c}(k)}K}} \right)} \right\rbrack} \leq {1 - M_{1}}} & (26)\end{matrix}$ wherein s(M) is the spectral radius of matrix M; letting$A = {{I - \left( {\frac{\rho{\Phi_{c}(k)}{{\hat{\Phi}}_{c}^{T}(k)}}{\lambda + {{\hat{\Phi}(k)}}^{2}} + {\beta{\Phi_{c}(k)}K}} \right)}}_{v}$and B=∥βΦ_(c)(k)K)∥_(v); from the conclusion of the spectral radius ofthe matrix, an any small positive number ε₁ exists, such that$\begin{matrix}{A \leq {{s\left\lbrack {I - \left( {\frac{\rho{\Phi_{c}(k)}{{\hat{\Phi}}_{c}^{T}(k)}}{\lambda + {{\hat{\Phi}(k)}}^{2}} + {\beta{\Phi_{c}(k)}K}} \right)} \right\rbrack} + \varepsilon_{1}} \leq {1 - M_{1} + \varepsilon_{1}} < 1} & (27)\end{matrix}$ wherein ∥M∥_(v) is the compatible norm of matrix M; βexists such that B satisfies the following inequality:1>1−A≤M ₁−ε₁ >B>0  (28) from formulas (16) and (28), obtaining:∥e(k+1)∥_(v) ≤A∥e(k)∥_(v) +B∥e(k−1)∥_(v)<(1−B)∥e(k)∥_(v)+B∥e(k−1)∥_(v)  (29) after transposition, obtaining:∥e(k+1)∥_(v) −∥e(k)∥_(v) <−B(∥e(k)∥_(v) −∥e(k−1)∥_(v))  (30) based onformula (30), discussing the form of e(k) from the following fouraspects: in a first case, when ∥e(k+1)∥_(v)>∥e(k)∥_(v) and∥e(k)∥_(v)>∥e(k−1)∥_(v), obtaining∥e(k+1)∥_(v) −∥e(k)∥_(v) >−B(∥e(k)∥_(v) −∥e(k−1)∥_(v))  (31) which isthe opposite of formula (30); therefore, this assumption does not exist;in a second case, when ∥e(k+1)∥_(v)>∥e(k)∥_(v) and∥e(k)∥_(v)<∥e(k−1)∥_(v) from formula (30), obtaining: $\begin{matrix}{\frac{{{e\left( {k + 1} \right)}}_{v} - {{e(k)}}_{v}}{{{e\left( {k - 1} \right)}}_{v} - {{e(k)}}_{v}} < B < 1} & (32)\end{matrix}$ i.e., the decrease of e(k) is larger than the increase inthree adjacent sampling points; and as a result, the overall trend isdecreasing under this situation; in a third case, when∥e(k+1)∥_(v)<∥e(k)∥_(v) and ∥e(k)∥_(v)<∥e(k−1)∥_(v), obtaining$\begin{matrix}{\frac{{{e\left( {k + 1} \right)}}_{v}}{{{e(k)}}_{v}} < {1{and}\frac{{{e(k)}}_{v}}{{{e\left( {k - 1} \right)}}_{v}}} < 1} & (33)\end{matrix}$ which satisfies formula (30), and e(k) has a decreasingtrend in this case; in a fourth case, when ∥e(k+1)∥_(v)<∥e(k)∥_(v) and∥e(k)∥_(v)>∥e(k−1)∥_(v), according to formula (30), this situation mayexist; two possibilities exist in the time of k+2 in detail: if∥e(k+2)∥_(v)>∥e(k+1)∥_(v) exists, obtaining the same conclusion as thesecond case; if ∥e(k+2)∥_(v)<∥e(k+1)∥_(v), obtaining the same conclusionas the third case; in short, e(k) still has a decreasing trend in thiscase; the above methods of proof are also applicable when f(k)=0; to sumup, the overall trend of error e(k) is decreasing; therefore, theconvergence of the error is proved; step D: applying the above controlalgorithm to control of an aero-engine model, and selecting threedifferent cases for result comparison to verify the effectiveness andsuperiority of the control algorithm; firstly, comparing the controleffects of MFAC+Kp, CFDL-MFAC and PID under the standard conditions toillustrate the effectiveness of an improved controller; and then,comparing the control effects at different heights and different delaysto illustrate the superiority of the controller.